Related papers: On Polya' Theorem in Several Complex Variables

Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in…

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of L-functions in families with the same…

We say that a subset of C^n is hypoconvex if its complement is the union of complex hyperplanes. Let D be the closed unit disk in C, T the unit circle. We prove two conjectures of Helton and Marshall. (See ``Frequency domain design and…

Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y),…

In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions…

Let $\mathcal F$ be either the set of all bounded holomorphic functions or the set of all $m$-homogeneous polynomials on the unit ball of $\ell\_r$. We give a systematic study of the sets of all $u\in\ell\_r$ for which the monomial…

In the present article, we investigate the univalence property of polyanalytic functions and $\log$-$\alpha$-analytic functions. First, by using a new idea, we prove an improved lemma and the coefficient estimates for bounded polyanalytic…

For a regular, compact, polynomially convex circled set K in C^2, we construct a sequence of pairs {P_n,Q_n} of homogeneous polynomials in two variables with deg P_n = deg Q_n = n such that the sets K_n: = {(z,w) \in C^2 : |P_n(z,w)| \leq…

In this paper we prove that if the polar decomposition of a symbol $f$ is truncated above, i.e., $f(re^{i\theta} )=\sum_{k=-\infty}^Ne^{ik\theta} f_k (r)$ where the $f_k$'s are radial functions, and if the associated Toeplitz operator $T_f$…

On $T \times G$, where $T$ is a compact real-analytic manifold and $G$ is a compact Lie group, we consider differential operators $P$ which are invariant by left translations on $G$ and are elliptic in $T$. Under a mild technical condition,…

We prove a conjecture concerning the third Hankel determinant, proposed in ``Anal. Math. Phys., https://doi.org/10.1007/s13324-021-00483-7", which states that $|H_3(1)|\leq 1/9$ is sharp for the class $\mathcal{S}_{\wp}^{*}=\{zf'(z)/f(z)…

Let $C$ be a subset of $\mathbb{R}^n$ (not necessarily convex), $f:C\to\mathbb{R}$ be a function, and $G:C\to\mathbb{R}^n$ be a uniformly continuous function, with modulus of continuity $\omega$. We provide a necessary and sufficient…

The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forelli's theorem on the complex analyticity of the functions that are: (1) $\mathcal{C}^\infty$…

In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\|f\|_{A_\alpha^{2\alpha}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $\alpha>1$. If the norms…

We give a combinatorial extension of the classical inequalities of Maclaurin about symmetric functions of several variables. We discuss two problems - one analytical and another combinatorial - and show that they are in some sense…

Let \( P(z) \) be a polynomial of degree \( n \) and $\alpha \in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_\alpha P(z) \) and is defined by $D_\alpha P(z): = nP(z) + (\alpha -z)P'(z)$. The polar derivative \( D_\alpha…

It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions…

Let $f$ be a holomorphic function on the unit disc, and $(S_{n_{k}})$ be a subsequence of its Taylor polynomials about $0$. It is shown that the nontangential limit of $f$ and lim$_{k\rightarrow \infty }S_{n_{k}}$ agree at almost all points…

Let $f(z) = \sum_{k=0}^\infty a_k z^k$ be an analytic function in a disk $D_R$ of radius $R>0$, and assume that $f$ is $p$-valent in $D_R$, i.e. it takes each value $c\in{\mathbb C}$ at most $p$ times in $D_R$. We consider its Borel…